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MCK's questions at Yahoo! Answers regarding an inexact ODE
- Thread starter MarkFL
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Hello MCK,
I would first write the ODE in differential form:
\(\displaystyle \left(y+3y^7 \right)dx+\left(-y^5-6x \right)dy=0\)
Next, let us compute:
\(\displaystyle \frac{\partial}{\partial y}\left(y+3y^7 \right)=1+21y^6\)
\(\displaystyle \frac{\partial}{\partial x}\left(-y^5-6x \right)=-6\)
Hence, we find the equation is inexact. So we look for an integrating factor. To do this, we consider:
\(\displaystyle \frac{(-6)-\left(1+21y^6 \right)}{y+3y^7}=-\frac{21y^6+7}{3y^7+y}\)
Since this is a function of just $y$, we find our special integrating factor with:
\(\displaystyle \mu(y)=\exp\left(\int -\frac{21y^6+7}{3y^7+y}\,dy \right)\)
To compute the integrating factor, we need to compute:
\(\displaystyle \int \frac{21y^6+7}{3y^7+y}\,dy\)
Let's write the integrand as:
\(\displaystyle \frac{21y^6+7}{3y^7+y}=\frac{21y^6+1}{3y^7+y}+ \frac{6}{3y^7+y}\)
The first term is easily integrated, and for the second term, let's use partial fraction decomposition to write:
\(\displaystyle \frac{6}{3y^7+y}=\frac{6}{y}-\frac{18y^5}{3y^6+1}\)
And so, putting all of this together, we find:
\(\displaystyle \mu(y)=e^{-\left(\ln\left|3y^7+y \right|+6\ln|y|-\ln\left|3y^6+1 \right| \right)}\)
Hence:
\(\displaystyle \mu(y)=\frac{3y^6+1}{y^6\left(3y^7+y \right)}=\frac{1}{y^7}\)
Multiplying the ODE by this factor, we obtain:
\(\displaystyle \left(y^{-6}+3 \right)dx+\left(-y^{-2}-6xy^{-7} \right)dy=0\)
Now, it is easy to see that we have an exact ODE. Hence:
\(\displaystyle \frac{\partial F}{\partial x}=y^{-6}+3\)
Integrating with respect to $x$, we find:
\(\displaystyle F(x,y)=xy^{-6}+3x+g(y)\)
Now, to determine $g(y)$, we differentiate with respect to $y$, recalling \(\displaystyle \frac{\partial F}{\partial x}=-y^{-2}-6xy^{-7}\):
\(\displaystyle -y^{-2}-6xy^{-7}=-6xy^{-7}+g'(y)\)
Thus, we find:
\(\displaystyle g'(y)=-y^{-2}\)
Hence, by integrating, we obtain:
\(\displaystyle g(y)=y^{-1}\)
And so we have:
\(\displaystyle F(x,y)=xy^{-6}+3x+y^{-1}\)
And so the solution to the ODE is given implicitly by:
\(\displaystyle xy^{-6}+3x+y^{-1}=C\)
edit: Hello MCK, I saw your message at Yahoo asking me to take a look at another of your questions posted there, however, Yahoo will not let me access the link, nor will it let me post a comment there. As an alternative, you could post the question here in our differential equations forum.
I would first write the ODE in differential form:
\(\displaystyle \left(y+3y^7 \right)dx+\left(-y^5-6x \right)dy=0\)
Next, let us compute:
\(\displaystyle \frac{\partial}{\partial y}\left(y+3y^7 \right)=1+21y^6\)
\(\displaystyle \frac{\partial}{\partial x}\left(-y^5-6x \right)=-6\)
Hence, we find the equation is inexact. So we look for an integrating factor. To do this, we consider:
\(\displaystyle \frac{(-6)-\left(1+21y^6 \right)}{y+3y^7}=-\frac{21y^6+7}{3y^7+y}\)
Since this is a function of just $y$, we find our special integrating factor with:
\(\displaystyle \mu(y)=\exp\left(\int -\frac{21y^6+7}{3y^7+y}\,dy \right)\)
To compute the integrating factor, we need to compute:
\(\displaystyle \int \frac{21y^6+7}{3y^7+y}\,dy\)
Let's write the integrand as:
\(\displaystyle \frac{21y^6+7}{3y^7+y}=\frac{21y^6+1}{3y^7+y}+ \frac{6}{3y^7+y}\)
The first term is easily integrated, and for the second term, let's use partial fraction decomposition to write:
\(\displaystyle \frac{6}{3y^7+y}=\frac{6}{y}-\frac{18y^5}{3y^6+1}\)
And so, putting all of this together, we find:
\(\displaystyle \mu(y)=e^{-\left(\ln\left|3y^7+y \right|+6\ln|y|-\ln\left|3y^6+1 \right| \right)}\)
Hence:
\(\displaystyle \mu(y)=\frac{3y^6+1}{y^6\left(3y^7+y \right)}=\frac{1}{y^7}\)
Multiplying the ODE by this factor, we obtain:
\(\displaystyle \left(y^{-6}+3 \right)dx+\left(-y^{-2}-6xy^{-7} \right)dy=0\)
Now, it is easy to see that we have an exact ODE. Hence:
\(\displaystyle \frac{\partial F}{\partial x}=y^{-6}+3\)
Integrating with respect to $x$, we find:
\(\displaystyle F(x,y)=xy^{-6}+3x+g(y)\)
Now, to determine $g(y)$, we differentiate with respect to $y$, recalling \(\displaystyle \frac{\partial F}{\partial x}=-y^{-2}-6xy^{-7}\):
\(\displaystyle -y^{-2}-6xy^{-7}=-6xy^{-7}+g'(y)\)
Thus, we find:
\(\displaystyle g'(y)=-y^{-2}\)
Hence, by integrating, we obtain:
\(\displaystyle g(y)=y^{-1}\)
And so we have:
\(\displaystyle F(x,y)=xy^{-6}+3x+y^{-1}\)
And so the solution to the ODE is given implicitly by:
\(\displaystyle xy^{-6}+3x+y^{-1}=C\)
edit: Hello MCK, I saw your message at Yahoo asking me to take a look at another of your questions posted there, however, Yahoo will not let me access the link, nor will it let me post a comment there. As an alternative, you could post the question here in our differential equations forum.